Suppose that $F(u, v) = \sum_{i}\sum_j u_i * v_i * C_{ij}$ is a bilinear matrix-valued function, where $C_{ij}$ are known matrix.

Is there a relatively easy way to factorize $F$ so that $u$ variables and $v$ variables are separated?

For example, find matrix $A_i$ and $B_j$ such that $F(u,v) = (\sum_i u_i * A_i) * (\sum_j v_j * B_j)$.

Suppose that $F(u, v) = \sum_{i}\sum_j u_i v_i C_{ij}$ is a bilinear matrix-valued function, where $C_{ij}$ are known matrices.

Is there a relatively easy way to factorize $F$ so that the $u$ and $v$ variables are separated? That is, to find matrices $A_i$ and $B_j$ such that $F(u,v) = (\sum_i u_i A_i)(\sum_j v_j B_j)$.